from sage.all import *
+def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None):
+ r"""
+ Generate a random rational convex polyhedral cone.
+
+ Lower and upper bounds may be provided for both the dimension of the
+ ambient space and the number of generating rays of the cone. If a
+ lower bound is left unspecified, it defaults to zero. Unspecified
+ upper bounds will be chosen randomly.
+
+ INPUT:
+
+ - ``min_dim`` (default: zero) -- A nonnegative integer representing the
+ minimum dimension of the ambient lattice.
+
+ - ``max_dim`` (default: random) -- A nonnegative integer representing
+ the maximum dimension of the ambient
+ lattice.
+
+ - ``min_rays`` (default: zero) -- A nonnegative integer representing the
+ minimum number of generating rays of the
+ cone.
+
+ - ``max_rays`` (default: random) -- A nonnegative integer representing the
+ maximum number of generating rays of the
+ cone.
+
+ OUTPUT:
+
+ A new, randomly generated cone.
+
+ EXAMPLES:
+
+ If we set the lower/upper bounds to zero, then our result is
+ predictable::
+
+ sage: random_cone(0,0,0,0)
+ 0-d cone in 0-d lattice N
+
+ In fact, as long as we ask for zero rays, we should be able to predict
+ the output when ``min_dim == max_dim``::
+
+ sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
+ 0-d cone in 4-d lattice N
+
+ TESTS:
+
+ It's hard to test the output of a random process, but we can at
+ least make sure that we get a cone back::
+
+ sage: from sage.geometry.cone import is_Cone # long time
+ sage: K = random_cone() # long time
+ sage: is_Cone(K) # long time
+ True
+
+ Ensure that an exception is raised when either lower bound is greater
+ than its respective upper bound::
+
+ sage: random_cone(min_dim=5, max_dim=2)
+ Traceback (most recent call last):
+ ...
+ ValueError: max_dim must be greater than or equal to min_dim.
+
+ sage: random_cone(min_rays=5, max_rays=2)
+ Traceback (most recent call last):
+ ...
+ ValueError: max_rays must be greater than or equal to min_rays.
+
+ """
+
+ # Catch obvious mistakes so that we can generate clear error
+ # messages.
+
+ if min_dim < 0:
+ raise ValueError('min_dim must be nonnegative.')
+
+ if min_rays < 0:
+ raise ValueError('min_rays must be nonnegative.')
+
+ if max_dim is not None:
+ if max_dim < 0:
+ raise ValueError('max_dim must be nonnegative.')
+ if (min_dim > max_dim):
+ raise ValueError('max_dim must be greater than or equal to min_dim.')
+
+ if max_rays is not None:
+ if max_rays < 0:
+ raise ValueError('max_rays must be nonnegative.')
+ if (min_rays > max_rays):
+ raise ValueError('max_rays must be greater than or equal to min_rays.')
+
+
+ def random_min_max(l,u):
+ r"""
+ We need to handle two cases for the upper bounds, and we need to do
+ the same thing for max_dim/max_rays. So we consolidate the logic here.
+ """
+ if u is None:
+ # The upper bound is unspecified; return a random integer
+ # in [l,infinity).
+ return l + ZZ.random_element().abs()
+ else:
+ # We have an upper bound, and it's greater than or equal
+ # to our lower bound. So we generate a random integer in
+ # [0,u-l], and then add it to l to get something in
+ # [l,u]. To understand the "+1", check the
+ # ZZ.random_element() docs.
+ return l + ZZ.random_element(u - l + 1)
+
+
+ d = random_min_max(min_dim, max_dim)
+ r = random_min_max(min_rays, max_rays)
+
+ L = ToricLattice(d)
+ rays = [L.random_element() for i in range(0,r)]
+
+ # The lattice parameter is required when no rays are given, so we
+ # pass it just in case.
+ return Cone(rays, lattice=L)
+
+
+def discrete_complementarity_set(K):
+ r"""
+ Compute the discrete complementarity set of this cone.
+
+ The complementarity set of this cone is the set of all orthogonal
+ pairs `(x,s)` such that `x` is in this cone, and `s` is in its
+ dual. The discrete complementarity set restricts `x` and `s` to be
+ generators of their respective cones.
+
+ OUTPUT:
+
+ A list of pairs `(x,s)` such that,
+
+ * `x` is in this cone.
+ * `x` is a generator of this cone.
+ * `s` is in this cone's dual.
+ * `s` is a generator of this cone's dual.
+ * `x` and `s` are orthogonal.
+
+ EXAMPLES:
+
+ The discrete complementarity set of the nonnegative orthant consists
+ of pairs of standard basis vectors::
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: discrete_complementarity_set(K)
+ [((1, 0), (0, 1)), ((0, 1), (1, 0))]
+
+ If the cone consists of a single ray, the second components of the
+ discrete complementarity set should generate the orthogonal
+ complement of that ray::
+
+ sage: K = Cone([(1,0)])
+ sage: discrete_complementarity_set(K)
+ [((1, 0), (0, 1)), ((1, 0), (0, -1))]
+ sage: K = Cone([(1,0,0)])
+ sage: discrete_complementarity_set(K)
+ [((1, 0, 0), (0, 1, 0)),
+ ((1, 0, 0), (0, -1, 0)),
+ ((1, 0, 0), (0, 0, 1)),
+ ((1, 0, 0), (0, 0, -1))]
+
+ When the cone is the entire space, its dual is the trivial cone, so
+ the discrete complementarity set is empty::
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: discrete_complementarity_set(K)
+ []
+
+ TESTS:
+
+ The complementarity set of the dual can be obtained by switching the
+ components of the complementarity set of the original cone::
+
+ sage: K1 = random_cone(max_dim=10, max_rays=10)
+ sage: K2 = K1.dual()
+ sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
+ sage: actual = discrete_complementarity_set(K1)
+ sage: actual == expected
+ True
+
+ """
+ V = K.lattice().vector_space()
+
+ # Convert the rays to vectors so that we can compute inner
+ # products.
+ xs = [V(x) for x in K.rays()]
+ ss = [V(s) for s in K.dual().rays()]
+
+ return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+
+
def lyapunov_rank(K):
r"""
Compute the Lyapunov (or bilinearity) rank of this cone.
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
+ TESTS:
+
+ The Lyapunov rank should be additive on a product of cones::
+
+ sage: K1 = random_cone(max_dim=10, max_rays=10)
+ sage: K2 = random_cone(max_dim=10, max_rays=10)
+ sage: K = K1.cartesian_product(K2)
+ sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+ True
+
+ The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
+ itself::
+
+ sage: K = random_cone(max_dim=10, max_rays=10)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
+
"""
V = K.lattice().vector_space()
- xs = [V(x) for x in K.rays()]
- ss = [V(s) for s in K.dual().rays()]
-
- # WARNING: This isn't really C(K), it only contains the pairs
- # (x,s) in C(K) where x,s are extreme in their respective cones.
- C_of_K = [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ C_of_K = discrete_complementarity_set(K)
- matrices = [x.column() * s.row() for (x,s) in C_of_K]
+ matrices = [x.tensor_product(s) for (x,s) in C_of_K]
# Sage doesn't think matrices are vectors, so we have to convert
# our matrices to vectors explicitly before we can figure out how
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